NCERT Class 10 Maths Lab Manual – Sum of Odd Natural Numbers
Objective
To verify that the sum of first n odd natural numbers, 1 +3 + 5+ … + (2n— 1) = n by paper activity.
Prerequisite Knowledge
 Natural numbers (i.e. counting numbers e.g., 1, 2, 3, 4, …)
 Odd natural numbers (Natural numbers which are not divisible by 2 i.e. 1, 3, 5, …)
 Even natural numbers (Natural numbers which are divisible by 2 i.e. 2, 4, 6,…)
 Formula to find the n ‘ term of an AP i.e.,
a_{n }= a+ (n—1) d
where a—> first term, d—> common difference, n —> no. of terms  For odd natural numbers 1,3,5,…, term is
a_{n} =1 + (n— 1). 2 = (2n— 1)  Area of squares.
Materials Required
Squared papers, sketch pens, pencil, a pair of scissors, geometry box, fevicol, white drawing sheets
Procedure

 Take a squared chart paper of size n units X n units (Take n— 9).
 Paste it on a white sheet.
 Colour the internal squares with different 9 colours as shown in the fig. (i).
 Take a squared chart paper of size n units X n units (Take n— 9).
4. Put a black dot on each of the internal coloured squares as shown in fig.(ii)
Observation
Result
The sum of first n odd natural numbers is n i.e., 1 + 3 + 5 + 7 + … + (2» —1) = n
Learning Outcome
Through this activity students are able to find that the sum of first n odd natural numbers n^{2 }i.e., \(\sum { (2n1) }\) =n^{2}
Activity Time
 Find out the sum of first fifteen odd natural numbers with the help of above activity and verify the result using the formula \(\sum { (2n1) }\) = n^{2}
 Find the sum of (11 + 13 + …+21) using the result of this activity.
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Viva Voce
Question 1:
What is the sum of first n natural numbers ?
Answer:
S_{n }= \(\frac { n(n+1) }{ 2 }\)
Question 2:
Give the formula for sum of first (« + 1) natural numbers.
Answer:
S_{n+1 }=\(\frac { (n+2)(n+1) }{ 2 }\)
Question 3:
What is the sum of first n multiples of 5 ?
Answer:
\(\frac { 5\times n(n+1) }{ 2 } \)
Question 4:
Whatisthesumof 2 + 6 + 10 + 14 + 18 + … + 10 terms ?
Answer:
200
[Hint: 2 [1 + 3 + 5 +… + 10 terms] =2 x 10^{2}= 200]
Question 5:
What is the #th term of 1 + 3 + 5 ?
Answer:
a_{n }=(2n— 1)
Question 6:
What is the n th term of 2 + 4 + 6 ?
Answer:
a_{n} = 2n
Question 7:
Give the formula for the sum to n terms of an AP.
Answer:
S_{n}= \(\frac { n }{ 2 } [2a+(na)d]\)
Question 8:
Define odd numbers and which is the first odd natural number.
Answer:
Numbers which are not divisible by 2 are known as odd numbers. 1 is the first odd natural number.
Multiple Choice Questions
Question 1:
Find n for \(\frac { n(n+1) }{ 2 }\) =55
(a) 11
(b) 10
(c) 11
(d) None of these
Question 2:
Using, \(\frac { n(n+1) }{ 2 }\) evaluate 1 + 3 + 5 + 7 + 9.
(a) 24
(b) 23
(c) 25
(d) 26
Question 3:
If n th term of an AP is (2n + 1), find the sum of first n terms of the AP.
(a) n(n+ 2)
(b) n(n 2)
(c) (n^{2}—2)
(d) None of these
Question 4:
For AP 5 + 4 + 3 + 2 + 1+ 0 + (1) + what should be its last term so that sum of all terms is zero ?
(a) 4
(b) 7
(c) 5
(d) None of these
Question 5:
In AP 16 + 12 + 8 + 4 + 0 + (4) + (8) + (12) +S7 is
(a) 28
(b) 16
(c) 20
(d) None of these
Question 6:
Common difference of an AP \(sqrt { 2 } ,\sqrt { 8 } ,\sqrt { 18 } ,\sqrt { 32 } \) is
(a) \(\sqrt { 3 } \)
(b) \(\sqrt { 2 } \)
(c) 2\(\sqrt { 3 } \)
(d) 4
Question 7:
First term of an AP is p and its common difference is q then its 10th term is
(a) 9p + q
(b) p9q
(c) p + 9q
(d) None of these
Question 8:
The sum of first npositive integer S_{n }is
(a) \(\frac { n(n1) }{ 2 } \)
(b) \(\frac { n(n+1) }{ 2 } \)
(c) \(\frac { n }{ 2 } [2a+(n1)d]\)
(d) a+(n1)d
Question 9:
20th term of the series 4, 7, 10, ……. is
(a) 61
(b) 60
(c) 59
(d) None of these
Question 10:
Choose the correct option : First four terms of the AP whose first term is —1 and common difference is \(\frac { 1 }{ 2 } \)
(a) 1 , –\(\frac { 1 }{ 2 } \) , \(\frac { 1 }{ 2 } \) , 0
(b) 1 , 0 ,\(\frac { 1 }{ 2 } \) , \(\frac { 1 }{ 2 } \)
(c) 1 , –\(\frac { 1 }{ 2 } \) , 0, \(\frac { 1 }{ 2 } \)
(d) 0 , –\(\frac { 1 }{ 2 } \) , \(\frac { 1 }{ 2 } \) , 1
Answers
1. (b)
2. (c)
3. (a)
4. (c)
5. (a)
6. (b)
7. (c)
8. (b)
9. (a)
10.(c)
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